Alonso had opened the scoring with an excellent free kick, before that effort was cancelled out by Michy Batshuayi's own goal moments after the Belgian had been introduced as a substitute.
However, two minutes from time, Alonso sent a shot under Hugo Lloris to earn the defending champions their first win of the new campaign.
Head coach Antonio Conte had handed Alvaro Morata his first start in a Chelsea shirt and the Spaniard should have scored in the fifth minute, but he instead sent a free header wide of the post from seven yards out after meeting Cesar Azpilicueta's cross.
Back came Spurs, with Harry Kane forcing Thibaut Courtois to spill his long-range effort and Dele Alli firing over from an acute angle, before Mousa Dembele forced a flying save out of Courtois with a swerving 25-yard strike.
Mauricio Pochettino's side were the better of the two teams but in the 24th minute, Chelsea went in front thanks to a sublime free kick from Alonso, who found the top corner of the net from 25 yards.
It was a stunning goal, but the remainder of the first half belonged to Spurs, who peppered Courtois's goal and will wonder how they were not at least on level terms at the break.
Courtois did well to deny Kane from close range, before the Spurs forward drilled a low effort just wide of the far post from 20 yards as he began to get the better of Antonio Rudiger.
Three minutes before half time, Kane rattled the far post after cutting inside onto his right foot, with Ben Davies also coming close to finding the corner with a deflected effort from the edge of the area which was tipped wide by Courtois.
After the restart, the game continued to be played at a high tempo but while Spurs were still bossing possession, especially in the Chelsea half of the pitch, chances were becoming less frequent as the visitors looked to find a way to nullify the threat.
However, the Chelsea backline were putting in block after block and after Spurs switched to a back four, it eventually freed up space for more opportunities to be created at the other end.
Morata saw a deflected effort go wide of the post, before Victor Moses fired a driven attempt narrowly over the crossbar after being allowed to advance into the penalty area.
It was then Willian's turn to go close as he lashed a low shot against the base of the post from 20 yards, but Chelsea failed to make their momentum pay and Spurs eventually equalised in the most unlikely of circumstances.
It appeared that there was only going to be one winner but with two minutes left, Chelsea regained their lead thanks to a second from Alonso, who fired the ball under Lloris from an acute angle after being teed up by Pedro.
Spurs attempted to find a second equaliser, but Chelsea efficiently saw out the remaining minutes to get their first points on the board for the season.